The invention relates to satellite positioning systems, such as the global positioning system, and more particularly to algorithms used to determine the position of a satellite in such a system, using in some applications information provided by other (non-satellite-based) positioning systems (such as a cellular network positioning system) and sensors.
In satellite positioning systems (such as the global positioning system or GPS), the position of a receiver (user) and its time offset from the system time (i.e. the correction to the receiver time at which the receiver is determined to be at the determined position) can be determined by using (pseudorange) measurements obtained from information (ephemerides and C/A-code phases) provided from at least four satellites. Such a determination can use satellite measurements at a particular instant of time, in what is called a single-point solution, a solution that in no way takes into account past information obtained from the satellites; any error in the measurements obtained from the satellites, including error from noise or multi-path, is reflected in such a single-point solution.
Filtering with a Kalman filter (or some modification of such a filter) can instead be used to enhance the quality of the receiver""s estimated track (by providing smoother, less noisy solutions than are provided by a single-point solution), and also to provide useable solutions in periods when satellite measurements are not available (because of poor signal conditions). The performance of any such filter is dependent on how the receiver""s motion is modeled in the filter. Usually, instead of using an ordinary Kalman filter, what is called an extended Kalman filter (EKF), which is a linearized form of Kalman filter, is used, because a standard Kalman filter assumes that the measurement update equations are linear, and for positioning problems the measurement update equations, which involve the pseudoranges, are nonlinear. For a (standard) Kalman filter to be used, there has to be a linear relationship between the measurement vector m and state vector s, such that m=Hxc2x7s, where H is some matrix. In GPS positioning, if the state vector is for example of the form [x y z t], where (x,y,z) indicates position and t represents clock bias, there is no such linear equation between pseudorange measurements and state. Instead, the ith component of the measurement vector (i.e. the pseudorange from the ith satellite), is given by
m(i)={square root over ((xixe2x88x92x)2+(yixe2x88x92y)2+(zixe2x88x92z)2)},
which is obviously not a linear relationship. In an EKF, to be able to still use a Kalman type filter in an application where such a nonlinear relationship exists, the nonlinear relationship is approximated by a linear relationship by forming a truncated Taylor series of the nonlinear equation and taking the first, linear term of the series. In practice, this means that the H matrix in the equation m=Hxc2x7s is approximated by the so-called Jacobian (known in the art) of the pseudorange equations.
Thus, in an EKF, a standard Kalman filter (for linear systems) is applied to nonlinear systems (with additive white noise) by continually updating a linearization around a previous state estimate, starting with an initial guess. In other words, linear Taylor series approximation (no nonlinear terms) of the system function at the previous state estimate is made, and a linear Taylor series approximation of the observation function at the corresponding predicted position. Such an approach yields a relatively simple and efficient algorithm for handling a nonlinear model, but convergence to a reasonable estimate depends to a great extent on the-accuracy of the initial guess at the desired position; the algorithm may not converge if the initial guess is poor or if disturbances to the motion are so large that linearization is inadequate to describe the system.
The prior art also teaches using what is called the interacting multiple model (IMM) solution, in which various motion models are assumed for the motion of the receiver (modules assuming slow turning, fast turning, slow accelerating, fast accelerating, and so on), and the outputs of the different models are combined based on weights that take into account how the predictions of the model agree with later measurements made on the basis of later information received from the satellites. In such an approach, each model (branch of the IMM solution) is implemented as an EKF.
The Kalman filter solution (usually an EKF solution), as noted, is in principal superior to a single-point solution in that it uses more information and provides a correspondingly more educated receiver position estimate. The IMM solution is in principal suitable for a wider range of applications than any single-model solution. But the prior art of GPS teaches using only an EKF for each model of an IMM solution.
The EKF is known to be inferior to an ordinary Kalman filter in performance, and the theoretical optimality results related to a Kalman filter do not apply to an EKF. In addition, using an EKF to implement each model of an IMM solution makes problematic using measurements based on information other than that provided by the positioning satellites, such as information from complementary positioning systems (e.g. cellular systems) and sensors including for example micro-electromechanical sensors such as inertial sensors (gyroscopes or accelerometers) and barometric altimeters. To integrate a measurement from a complementary positioning system into an EKF solution requires fusing another measurement to the state vector via some new nonlinear relationship which must be linearized and is therefore computationally costly.
The prior art also teaches a two-stage solution, depicted in FIG. 1, in which a single-point solution (versus a Kalman filter type or predictive filter solution which takes into account past measurement points) is used based on pseudorange measurements, and the single-point solution is then provided to a single Kalman filter. However, such an approach does not take advantage of the wider range of applicability of a solution based on the IMM.
What is needed, in order to provide a solution that is suitable in a wide range of applications and that provides reasonable estimates of a receiver""s position even in poor signaling conditions, is an IMM positioning solution that does not suffer from the defects of one that uses an EKF to implement each model. Ideally, the sought-after IMM positioning solution would require less calculations than an EKF-based IMM solution, and would allow for using information from complementary positioning systems and sensors in a more straightforward manner.
Accordingly, the present invention provides a method, a corresponding apparatus and a corresponding system for determining a dynamical quantity of a receiver of signals conveying information useful in estimating the dynamical quantity, the method including the steps of: providing a single-point solution, by solving for the dynamical quantity of the receiver using a single-point solution having as an input the information useful in estimating the dynamical quantity being determined; providing a plurality of filter solutions (such as predictive filter solutions) each assuming a different motion model for the receiver; and combining the plurality of filter solutions to provide a first value of the dynamical quantity based on weights that take into account the likelihood of the suitability of each motion model, with the likelihood determined on the basis of agreement of the first value of the dynamical quantity compared with a second value of the dynamical quantity as indicated by a single-point solution.
In a further aspect of the invention, each predictive filter is an ordinary Kalman filter for which no linearizing of the measurement update equations of the Kalman filter is performed.
In another further aspect of the invention, the dynamical quantity being determined is a quantity comprising one or more unknowns with respect to the receiver selected from the set consisting of: clock bias, position, clock drift, velocity, clock jerk, and acceleration.
A system according to the invention, includes in some applications not only a receiver with components according to the invention, but also includes an external computing facility, not hosted by the receiver, and coupled to the receiver via wireless communications enabled by a cellular communication system, wherein at least some of the computation of either the single-point solution or the predictive filters is performed in the external computing facility and communicated to the receiver via wireless communication. In some other applications, a system, according to the invention, for determining a dynamical quantity of a receiver of signals conveying information useful in estimating the dynamical quantity includes not only a receiver with components according to the invention, but also at least one satellite for providing the signals conveying information useful in estimating the dynamical quantity.
There are several advantages in the use of a two-stage IMM solution as opposed to a single-stage (EKF-based) IMM solution. First, the filters used are linear Kalman filters, and so are theoretically optimal and also have a predictable performance. One of the most significant problems associated with the EKF (whether or not used to implement the models of an IMM) is that the measurement equation linearization has to be carried out with respect to some particular solution point. In navigation applications, the particular point is the predicted position and time offset estimate. If the position estimate is badly mis-predicted, so is the linearization, leading to a diverging sequence of outputs from the EKF. A linear Kalman filter does not suffer from such a problem; as stated in GPS Positioning, Filtering and Integration by J. Chaffee, J. Abel, and B. McQuiston in IEEE Aerospace and Electronics Conference 1993, which is hereby incorporated by reference in its entirety as background, a closed-form solution in the first stage is a near sufficient statistic in that it conveys essentially the same information as a set of raw pseudorange measurements, summarizing the information contained in the raw measurements, and a two-stage filter using a linear Kalman filter actually preserves more of the measurement information than does an EKF.
A second advantage is that the two-stage approach of the invention allows using an extremely simple relationship between the measurement vector and the state vector, namely the identity relationship, s=m, since the xe2x80x9cmeasurementsxe2x80x9d in the invention are actually state vector (single-point) solutions, so that the H matrix is the identity matrix, which simplifies all of the equations of the Kalman filters used in the invention, and many costly matrix operations are avoided, especially if sequential measurement processing is used. Altogether, the linearity and simplicity of the Kalman filters lead to substantial savings in computational costs.
Another benefit worth mentioning is that with the invention it is possible to choose a first-stage pre-filter from among different single-point solution algorithms. For example, when only three satellites are seen, but the receiver knows its altitude, Phatak""s point-solution, described in Position Fix fromitThree GPS Satellites and Altitude: A Direct Method by M. Phatak, M. Chansarkar, and S. Kohli in IEEE Transactions on Aerospace and Electronic Systems, Vol. 35 (1) January 1999 (which is hereby incorporated by reference in its entirety as background) can be applied, and when four or more satellites are seen; Bancroft""s point solution can be used.
Yet another advantage is that in the present invention, an increase in the number of measurements does not significantly add to the cost in filtering computations. It is in general advantageous to use as many measurements as possible, but in case of a pure filter solution, the number of measurements increases the processing load of each filter. In the two-stage solution of the invention, however, the number of measurements affects only the first stage (the single-point solution), so that the cost of the second stage (the filters) is constant. Thus, instead of adding the computational load in all the parallel filters, the added information burdens the part of the algorithm that is performed only once at each filtering cycle.
Finally, additional measurements from other positioning systems such as network-based systems or sensors such as inertial sensors can be readily used by the invention because of the simple form of the filter input xe2x80x9cmeasurementxe2x80x9d given by the first-stage single-point solution.